Before we jump into understanding how quantum computing works from the ground up, we need to take a little time to see how things are done classically. This is not only for the sake of comparison. The future, I believe, will be a hybrid of classical and quantum computers.
The best way to learn about something is to start with basic principles and then work your way up. That way, you know how to reason about it and don’t rely on rote memorization or faulty analogies.
The first part covers the mathematics you need to understand quantum computing concepts. While we will ultimately be operating in very large dimensions and using complex numbers, you can gain a lot of insight from what happens in traditional 2D and 3D.
Chapter 1 – Why Quantum Computing
In the first chapter, we ask the most basic question that applies to this book: why quantum computing? Why do we care? In what ways will our lives change? What are the use cases to which we hope to apply quantum computing and see a significant improvement? What do we even mean by “significant improvement”?
Chapter 2 – They’re Not Old, They’re Classics
Classical computers are pervasive, but relatively few people know what’s inside them and how they work. To contrast them later with quantum computers, we look at the basics along with the reasons why they have problems doing some kinds of calculations. I introduce the simple notion of a bit, a single 0 or 1, but show that working with many bits can eventually give you all the software you use today.
Chapter 3 – More Numbers Than You Can Imagine
The numbers people use every day are called real numbers. Included in these are integers, rational numbers, and irrational numbers. Other kinds of numbers and structures have many of the same algebraic properties. We look at these to lay the groundwork for understanding the “compute” part of what a quantum computer does.
Chapter 4 – Planes and Circles and Spheres, Oh My
From algebra, we move to geometry and relate the two. What is a circle, really, and what does it have in common with a sphere when we move from two to three dimensions? Trigonometry becomes more obvious, though that is not a legally binding statement. What you thought of as a plane becomes the basis for understanding complex numbers, which are key to the definition of quantum bits, usually known as qubits.
Chapter 5 – Dimensions
After laying the algebraic and geometric groundwork, we move beyond the familiar two- and three-dimensional world. Vector spaces generalize to many dimensions and are essential for understanding the exponential power that quantum computers can harness. What can you do when working in many dimensions, and how should you think about such operations? This extra elbow room comes into play when we consider how quantum computing might augment AI.
Chapter 6 – What Do You Mean “Probably”?
“God does not play dice with the universe,” said Albert Einstein. Einstein, Albert
This was not a religious statement but rather an expression of his lack of comfort with the idea that randomness and probability play a role in how nature operates. Well, he didn’t get that quite right. Quantum mechanics, the deep and often mysterious part of physics on which quantum computing is based, very much has probability at its core. Therefore, we cover the fundamentals of probability to aid your understanding of quantum processes and behavior.
The next part is the core of how quantum computing really works. We look at quantum bits—qubits—singly and together, and then create circuits that implement algorithms. Much of this is the ideal case when we have fault-tolerant, error-corrected qubits. When we build quantum computers, we must deal with the physical realities of noise and the need to reduce errors.
Chapter 7 – One Qubit
At this point, we can finally talk about qubits in a nontrivial manner. We look at both the vector and Bloch sphere representations of the quantum states of qubits. We define superposition, which explains the common cliché about a qubit being “zero and one at the same time.”
Chapter 8 – Two Qubits, Three
With two qubits, we need more math, and so we introduce the notion of the tensor product. This allows us to explain entanglement, which Einstein called “spooky action at a distance.” Entanglement tightly correlates two qubits so that they no longer act independently. With superposition, entanglement gives rise to the very large spaces in which quantum computations can operate. spooky action at a distance qubit$entanglement
Chapter 9 – Wiring Up the Circuits
Given a set of qubits, how do you manipulate them to solve problems or perform calculations? The answer is you build circuits for them out of gates that correspond to reversible operations. For now, think about the classical term “circuit board.” I use the quantum analog of circuits to implement algorithms, the recipes computers use for accomplishing tasks.
Chapter 10 – From Circuits to Algorithms
With several simple algorithms discussed and understood, we turn to more complicated ones that fit together to give us Peter Shor’s 1995 fast integer factoring algorithm. This chapter’s math is more extensive, but we have everything we need from previous discussions.
Chapter 11 – Getting Physical
When you build a physical qubit, it doesn’t behave exactly like the math and textbooks say it should. There are errors, and they may come from noise in the environment of the quantum system. I don’t mean someone yelling or playing loud music, I mean fluctuating temperatures, radiation, vibration, and so on. We look at several factors you must consider when you build a quantum computer, introduce Quantum Volume as a whole-system metric of the performance of your system, and conclude with a discussion of the most famous quantum feline.
The final part of this book looks at more advanced topics that may require additional physics or machine learning background.
Chapter 12 – Considering NISQ Algorithms
Noisy Intermediate-Scale Quantum, or “NISQ,” computers have qubits that are not fully fault-tolerant and error-corrected. Decoherence, together with initialization, gate, and measurement errors, make calculations even more unpredictable than probability would indicate. Are there algorithms that use small quantum circuits intermixed with classical methods to approximate solutions to exponentially hard problems? NISQ Noisy Intermediate-Scale Quantum era
Chapter 13 – Introduction to Quantum Machine Learning
Many researchers have begun to look at whether quantum methods can augment AI and machine learning. This research has not yet shown an advantage over purely classical methods, but we survey several topics, such as neural networks and kernel methods, to understand the approaches and the issues.
Chapter 14 – Questions about the Future
This book concludes with a chapter that moves beyond today by asking motivating questions to determine how “quantum-ready” you are to work with this new technology today and as it evolves.
